Solve 1/4z^2-1=a/2z-1-b/2z+1

$$\frac { 1 } { 4 z ^ { 2 } - 1 } = \frac { a } { 2 z - 1 } - \frac { b } { 2 z + 1 }$$

Solve for a

$$a=-\frac{-2bz+b-1}{2z+1}$$

$$|z|\neq \frac{1}{2}$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(2z-1\right)\left(2z+1\right), the least common multiple of 4z^{2}-1,2z-1,2z+1.

$$1=\left(2z+1\right)a-\left(2z-1\right)b$$

Use the distributive property to multiply 2z+1 by a.

$$1=2za+a-\left(2z-1\right)b$$

Use the distributive property to multiply 2z-1 by b.

$$1=2za+a-\left(2zb-b\right)$$

To find the opposite of 2zb-b, find the opposite of each term.

$$1=2za+a-2zb+b$$

Swap sides so that all variable terms are on the left hand side.

$$2za+a-2zb+b=1$$

Add 2zb to both sides.

$$2za+a+b=1+2zb$$

Subtract b from both sides.

$$2za+a=1+2zb-b$$

Combine all terms containing a.

$$\left(2z+1\right)a=1+2zb-b$$

The equation is in standard form.

$$\left(2z+1\right)a=2bz-b+1$$

Divide both sides by 2z+1.

$$\frac{\left(2z+1\right)a}{2z+1}=\frac{2bz-b+1}{2z+1}$$

Dividing by 2z+1 undoes the multiplication by 2z+1.

$$a=\frac{2bz-b+1}{2z+1}$$

Solve for b

$$b=-\frac{1-a-2az}{2z-1}$$

$$|z|\neq \frac{1}{2}$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(2z-1\right)\left(2z+1\right), the least common multiple of 4z^{2}-1,2z-1,2z+1.

$$1=\left(2z+1\right)a-\left(2z-1\right)b$$

Use the distributive property to multiply 2z+1 by a.

$$1=2za+a-\left(2z-1\right)b$$

Use the distributive property to multiply 2z-1 by b.

$$1=2za+a-\left(2zb-b\right)$$

To find the opposite of 2zb-b, find the opposite of each term.

$$1=2za+a-2zb+b$$

Swap sides so that all variable terms are on the left hand side.

$$2za+a-2zb+b=1$$

Subtract 2za from both sides.

$$a-2zb+b=1-2za$$

Subtract a from both sides.

$$-2zb+b=1-2za-a$$

Combine all terms containing b.

$$\left(-2z+1\right)b=1-2za-a$$

The equation is in standard form.

$$\left(1-2z\right)b=1-a-2az$$

Divide both sides by -2z+1.

$$\frac{\left(1-2z\right)b}{1-2z}=\frac{1-a-2az}{1-2z}$$

Dividing by -2z+1 undoes the multiplication by -2z+1.

$$b=\frac{1-a-2az}{1-2z}$$